A "change of basis" is an action performed in linear algebra, whereby a change in fundamental structure yields an entirely new viewpoint. This blog began as a record of a pedagogical change of basis for me, and continues as an ongoing account of my thoughts as I design and direct courses in mathematics at the University of North Carolina, Asheville.

Thursday, February 24, 2011

I got to campus by about 7:30 so I could take care of some bureaucratic matters and get ready for the "reboot" of my MATH 179 Ethnomathematics course. My intent today was to transition smoothly from the textbook by Marcia Ascher we've been using to guide the course so far to Stanislas Dehaene's (about the relative merits of these books I blogged yesterday).

That we did: we spent about half an hour talking about least common multiples, primes, and relative primeness. These concepts all come up in understanding Mayan calendar. Though some of the students were shaky with the concepts at first, they all seemed willing to engage them, and moreover the concepts are more concrete than many of those we've worked with so far this semester (like Marshall Islander stick charts and months based on Jupiter's moons). This concreteness helped students get a grip. We even dished a bit about Gauss's formula for the sum of the first n natural numbers and the conjectural infinitude of twin prime pairs.

The smooth slide into Dehaene that I'd hoped for came with an exercise I'd put together to test one of the points Dehaene makes in Chapter 4 of his text (which students are to read for next Tuesday): people who grow up with either English (13 of the 14 students present today) or French (the remaining one of the 14) as their native language have difficulty in remembering any series of more than seven or eight randomly generated digits shown to them 20 seconds previously. All 14 students were successful at remembering five, six, or seven digits, and at eight people started to falter: two students failed to get all eight, and several more failed to get nine. No one remembered ten correctly.

I felt renewed energy in class today. I'm looking forward to seeing what the students think about the reading for next week.

From Ethnomathematics I went straight to my colleague Louise's LANG 120 (our first-year composition) course, where I was guest-lecturing on the subject of writing in mathematics. Louise hoped that I could expose the students to some of the conventions of mathematical writing, partly in order to demonstrate that many of those conventions, and the criteria by means of which the quality of writing can be measured, are not all that different from those of academic writing in general. I think I succeeded in this to some extent. It was a lot of fun! It seemed like a good class, with some very outgoing and eager students.

In Calc II I (boy, that looks weird) tried out a new format for the students' quiz. Rather than offering a collaborative quiz (as the last few have been), and rather than making the quiz a fully solitary activity, I gave the students a brief "consultation period" in the middle of the exercise. The students had several minutes to get a start on solving the problem posed to them, and then I allowed them roughly two minutes to confer with one another in whatever way they wanted to, sharing ideas, checking their answers against each others', etc. This period over, they returned to solo work before submitting their quizzes. My theory was that this format would help students in much the same way "think/pair/share" exercises help them: the initial brainstorming and groundwork is done on a solo basis, but then conference with their colleagues helps to refine their initial thoughts as they take shape.

I can't tell how people felt about this, or if it had a noticeable effect on performance on the quiz. (The students did pretty well, making, for the most part, the errors I'd expect them to make. Nothing out of the ordinary.) Only one student offered feedback on the format on the quiz itself, indicating that she felt it threw her off more than it helped her.

If you're in my Calc II class, how do you think it went? If you're not in my Calc II class, how do you think you'd respond to this activity? Would it help you? Hinder you? What up? Feedback, please!

Wednesday, February 23, 2011

A highlight: another meeting of the minds in my office, involving three 280 students who'd stopped by after class for a debriefing on their recently-returned homework assignments. I'd mentioned in class just an hour before how beneficial others had found such debriefings, and these three had come hoping to reap the same benefits.

The meeting was wonderful. Sibyl, Nigel, and Quark joined me in a rough circle in front of my desk, and we went through the homework together, one problem at a time. Sometimes it took no more than a minute to iron out the wrinkles they'd worked themselves into, and other times we spent ten or twelve minutes puzzling through a problem's subtleties. They fed my wisdom and of their own, which was often richer. It's good for the students to see that others often have the same struggles they do (more than once they made the same kind of mistakes on the same problems), and it's good for them to hear their peers' explanations (often more lucid than my own). We ended the meeting with my giving them a few impromptu inductive exercises for practice with the technique. (Prove: n nonparallel lines in the plane determine n(n+1)/2 + 1 regions, bounded or unbounded.)

I left our meeting invigorated. I love this aspect of my job.

A second highlight: I realized this afternoon, while rereading Stanislas Dehaene in preparation for MATH 179 tomorrow, why it is I've had such a hard time getting into the mathematical aspects of my Ethnomathematics course this semester: the book, by Marcia Ascher's Mathematics elsewhere: An exploration of ideas across cultures, is awful, at least for my course. It's a bad fit. It's boring, condescending, pedantic, preachy, and is aimed at entirely the wrong level. (It's too dry to intrigue most math majors but too mathematically sophisticated to appeal to non-mathematicians.)

By comparison, Dehaene's book, though not strictly speaking about ethnomathematics (it deals more with the psychology and neuroscience of math and math learning), is intriguing, engaging, and written in a manner that's neither condescending nor confusing. I think the students will find it much more interesting than Ascher's text. We'll start drawing from it tomorrow and next week.

It's been a good day; I'm looking forward to another one tomorrow.

A closing note: my thanks to all of you who offered me support after my day of frustration yesterday. I'm sure regular readers will recognize that blogging is a cathartic exercise for me (need evidence? Check out the "anxiety" and "bitching" tags on the right!), and I often feel tremendously better after writing.

Tuesday, February 22, 2011

Late this morning I stopped for about a minute on the landing midway between the ground and the second floor in the stairwell at the far east end of Robinson Hall. I leaned against the wall and let the cold yellow brick rub into my back. I breathed deeply, hung my head, and closed my eyes. For a few minutes then I felt like crying. Then it passed.

I've often felt overwhelmed lately, what with four preps (i.e., teaching four different courses this term), a book draft due in dwindlingly few weeks, and an ever-growing glut of REU applications piling up in my in-box...the budget situation doesn't help, and I'm increasingly annoyed by the website migration, committee governance responsibilities, and the almost farcically bureaucratic assessment-related demands on my time brought about by the university's upcoming reaccreditation site visit. (Do I really need to write another impact statement? How long before we've simply assessed ourselves to death?)

I think perhaps what's most stressing and distressing to me lately (and may even be the root cause of the passionlessness I've lately expressed for my discipline?) is the distance I seem to have drifted from the aspect of my job I enjoy the most: hands-on, one-on-one work with my students. I don't feel like I get as much of that work as I used to, but when I do it's effect is as if godsent.

I had a few refreshing moments today. Kip, Kamryn, and Miriam all took me up on the offer I made my 280 students yesterday: I encouraged them each to come by and conference with me about the comments I made on their most recent homework assignments. Such conferences give me a chance to contextualize the issues I found in their writing, and to offer them more substantial guidance as they develop as writers and provers. The conferences also give them a chance to ask questions, to probe more deeply the problems I posed to them, and to get help on putting together new proofs that build on the old ones they've put in place before.

Every one of the conferences I had today was a helpful one, at least for me. Every one helped me normalize my expectations with those of the student I met; every one helped me better understand each student's weaknesses and strengths. (For instance, one fears being a "fraud" or an "impostor" because her strongest proof mimicked one we'd done together as a class.) I now feel as though I know each student who met with me just a little bit better than I know any of their classmates.

Moreover, as I hinted above, these conferences gave me a much-needed connection to the most meaningful aspect of my work I've too often felt missing lately. Meeting with these 280 folks, and strolling through the Math Lab and helping out with the odd trig-sub integral, really helped me get over my morning's mini-breakdown. I hope I was able to help a few other folks to wrestle down a few of their own personal demons.

We're all stressed, I can tell. But we're none of us alone. Let's all lean on each other. I'll slay your dragons if you slay mine. Together we'll make it through.

The students have spoken. It's clear from the feedback I got at the end of class today that I've gotten a bit off-topic in MATH 179 for the past couple of weeks as we've undertaken a more in-depth analysis of the university's structure and the impact that budget cuts will have on that structure.

The students definitely want to get some more math into the course, and we'll begin doing that on Thursday when we look at the Mayan calendar more closely. I'm also thinking of bringing in some reading from Stanislas Dehaene's The number sense: How the mind creates mathematics. I found this book fascinating reading a few years back, and I think it'll start some interesting conversations. Ditto Anthony Aveni's Uncommon sense: Understanding nature's truths across time and culture.

Breaking news: the administration has asked that our department cut six core mathematics courses from our fall schedule (including Calc I, Precalc, and STAT 185, our introductory non-calc-based statistics course) in order that we can free up people to teach in the Humanities program and LSIC courses (like the 179 I'm teaching right now).

Leaving aside the question of the relative value of the courses in question here (how much do we as a university value the Humanities program and our ILS Colloquia?), this is pure asininity, plain and simple. It's wrongheaded.

The effects would be staggering: already-huge sections of the classes mentioned above would get larger. We're already teaching with caps of 32 in Calc I and Precalc (effectively pushing numbers up to 35 or so for these courses when flexible instructors let a few folks in over the line) and 28 in STAT 185. The classrooms we're given barely hold those numbers, if they hold them at all. Cutting even a single section of Calc I would push numbers in the remaining sections up by about 5-6 students. We're now talking 40-person sections of Calc I and Precalc, and perhaps 35-person sections of STAT 185 (computer shortage is an issue there).

It is impossible to provide the meaningful student-centered instruction expected at a liberal arts institution in a course with that many students. My suspicion is that the powers that be who are mandating this move are operating under the impression that mathematics instruction is unidirectional and purely lecture-based. They run the danger of turning mathematics instruction at UNCA into something akin to the large-lecture methods employed at our giant sister schools down the road. Despite innovations like classroom response systems, this arrangement's still got nothing on one-on-one interaction between teacher and student.

Even from a simple human resources standpoint this move is silly: it takes faculty away from courses capped at 32 to courses capped at 22, at a time when we need to make the most of every faculty member's time and energy. Of course, in most departments this would be an even trade-off, since (according to a senior colleague down the hall who ran the numbers this morning) our department teaches more 30-plus-student sections than the rest of the university combined.

That same colleague and my department chair are now at a meaning at which they hope to try to reverse this request. I'll update later.

Thoughts?

UPDATE: after further negotiations this morning, we've managed to keep Calc I and Precalc intact, compromising by scrapping one section of STAT 185 and taking on two sections of Humanities. We've also pulled one of the longtime administrators (a former regular member of our department before leaving us mortals behind) to teach one section of our Nature of Mathematics (the name for our "general education" math course). I'm actually very heartened by this gesture, for I know how busy this woman is anyway. I'm grateful to her.

Thursday, February 17, 2011

I had the Calc II students tweeting again today, this time about the appropriate way to solve trig integrals involving sines and cosines. In 140 characters or less, how much can you say about solving such integrals? Below are some of the strongest responses. (By the way, "EFTI" means "Everyone's Favorite Trig Identity," my term for the identity sin2(θ) + cos2(θ) = 1.)

"If you see an odd # of cos(x), u=sin(x).If you see an odd # of sin(x), u=cos(x).If they are both even then throw your things and cry:("

"If there is an odd number of cos(x),letu=sin(x). If there is an odd number of sin(x) let u=cos(x).Then use trig identity if needed."

"To solve integrals with sin & cos, u=sinif ur cos is odd. u=cos if ur sin is odd. If both are even: weep. And Patrick spake it thus."

"Use EFTI!If the trigS has an odd and an even, make u=the even term.If the exponent is higher than 2,it may be easiest to seperate it first!"

"Integral with sin&cos: u sub the non-odd and sub the remainder with trig identity, then simplify. The rest should be easy."

"To solve intergral w/ sin & cos/if you see odd # of cos u=sin if you see odd # of sin u=cos/think of how to split them upsoit works out right"

"when u hav sine&cosine,ur u is the nonodd one.if both r odd u can choose either then use usub and EFTI to solve the trigintegral.its so fun!"

"In general: If the integral has an odd # of sins or cos, make the opposite one the u in the u-substitution set up.(Be careful of evens!)"

"If the powers are even/odd, use the even one as u. Iff there are extra of the odds, use EFTI. If both are odd, use the higher power as u."

I should mention that the overall quality of these tweets was stronger than that of the last set. Words were selected more carefully and fewer students felt the need to forgo spaces to say what they needed to in the room provided. I'm not sure if this improvement is a consequence of a more easily-summarized topic or a growing acquaintance with the genre. Maybe a little of both?

We had another round of "Everything you always wanted to know about UNCA (but were afraid to ask)" in MATH 179 today. It came at the end of the class (after further discussion of Jupiter's moons and an overview of the Integrative Liberal Studies program at UNC Asheville), so we didn't have enough time to address every question asked, but we got a few in. We'll finish the rest on Tuesday.

I'm impressed by the students' candor and curiosity, and I'm equally impressed by the students' knowledge of campus functioning. Some of these kids are definitely up with what's going down.

Questions?

What is the cause for the recent budget cuts?

What are the advantages/disadvantages of declaring a major?

If 179 is one of the writing intensives, what are the other 2?

What all will the new health and wellness center have to offer?

Will any majors be dropped with the budget cuts?

Where do I find info on clubs?

What are the rules for the disc golf course? And how many holes is it?

How much of my tuition is going toward building the community/campus gym?

Where do you apply for a learning disability? (I write on the side WITHOUT lines, I'm such a rebel!)

How many hours are required to graduate?

When will the construction for Governors Village be finished?

If I took a Spanish at my other school and it counted as both Spanish requirements if you passed and I passed then I scored below that level on the placement test what happens?

We got around to the first four of these today. I have to admit to a little relief that some of the others have yet to come up: I'm going to have to look up the answers to a few of them!

Wednesday, February 16, 2011

Strengths: I think I did a helluva job distilling the history of the WAC/WID/WTL movements down to about 5-10 pages. I think I did a good job of motivating WAC, WID, and WTL in the first place.

Weaknesses: In moving the section on student resistance from its original location (at the end of Chapter 2, on the writing process), the inter-section transitions at this new chapter's end may be a bit awkward.

We'll see. I've just sent Libby this draft; I'll eagerly await her feedback. She promises to get some reading in en route to Tuscaloosa, home to this year's Southern Writing Center Association conference. I wish I were going! Odd that this year Bama's playing host both to SWCA and to the Southeastern Sectional MAA meeting, coming up in a little over a month.

Now...to dink around on Facebook for a little while before beddie-bye. Au revoir!

Tuesday, February 15, 2011

This morning in Ethnomathematics we finally had a chance to spend some time trying to reconcile the "lunar" and "solar" calendars one might use were one stationed on Jupiter and decided to use the largest of the Galilean satellites, Ganymede, to reckon your "months."

A Ganymedean month (the time it takes to make one transit around Jupiter) is roughly 7.15 (Earth) days; meanwhile, Jupiter's tropical year (the time it takes the planet to make one transit around the sun) is 4331.57 (Earth days). Just as on Earth, where a tropical year doesn't contain an even number of lunar months, resulting in a "drift" between lunar-based and solar-based calendars, there are an uneven number of Ganymedean months in a Jovian year. What to do?

After a bit of numerical piddling and fiddling, we figured out two reasonably accurate ways of making the numbers jibe. Both rely on the fact that each Jovian year we have a "remainder" of 5.82 days that don't quite make up a full Ganymedean month. If we let 11 Jovian years pass, we've saved up 64.02 spare days...this figure is very close to 9 full Ganymedean months, which give us 64.35 days. Therefore, if we add 9 months every 11 years (which can be done in some systematic fashion, much as is done with the Jewish lunar calendar), we've add only 0.33 extra days. This overage is tantalizingly close to 1/3...so why not simply take away one day every three 11-year cycles? This day too can be chosen systematically, removed from the middle of the 17th year of the 33-year cycle it corresponds to, for instance.

Neat!

I'll leave it to my readers (I'm sure your curiosity is now piqued) to puzzle through the details of the other solution we arrived at, which had much the same flavor and involved slightly more frequent adjustments.

Fun stuff...now if we can only figure out how to make these systems fit nicely with the 365.24-day Earth tropical year, which some of our Jovian emigrés still insist on using...

As I write this the first MATH 280 "editing party" of the semester is going on in the Math Lab. Six of the class's 36 students showed up to look over one anothers' submissions for Chapter 1 of the textbook.

I'm eager to see how well the students receive this assignment this term, especially in such a large class!

Monday, February 14, 2011

At the end of class today I asked my Calc II students to write Twitter-style tweets in which they describe how to use the method of integration by parts. To help them stay within the 140-character limit of the genre, I provided them with graph paper on which I'd blocked out boxes containing 140 squares apiece.

The results?

They found it hard to say all that they wanted to say, so they had to focus on the issues most central to the method. (This, of course, was the point of the exercise.) Some tried to save space by eliminating spaces between words or by eliminating vowels. The former resulted in text that was still readable; the latter, I would argue, did not. Compare:

Some of the best were short and to the point, and even managed to get in some guidance about how to choose u and dv: "Sudv=uv-SvduumustgetsimpledvmustbeS.use formula above solve" was far terser than it needed to be; throw in a few spaces to make it more readable, and expand on what "mustbeS" surely means:

"Sudv=uv-Svdu. u must get simple. dv must be integrable. use formula above. solve."

is still well within the 140-character limit.

Clearer, but not quite as complete (it makes no mention of dv), and still pretty solid:

"pick a portion of the original integral that will get simpler when derived and set it to u, then do int(f(x))=uv-int(vdu)."

Several others were nearly as good, but used slightly awkward notation or omitted critical pieces of the formula. From what I can tell, the students get the gist of the process, but may still be a little iffy on the details.

Practice, people, practice! This gets much easier after you've done a few of them. If you want to go over a few examples together, please come on by. I'd love to work on a few with you. You'll get there, I promise.

Last week I asked my MATH 179 students to reflect on what "liberal arts" meant to them. They were to write an ungraded page or two about what they thought it is that distinguishes a liberal arts institution from one of a different sort. Their responses to this exercise were uniformly astute. They've got a good grasp of what a liberal arts education entails already, so my goal for the next of the semester need not be so much to introduce them to this educational philosophy so much as it will be to help them investigate its subtleties and implications.

Many focused on the relatively intimate learning environment at liberal arts colleges, bolstered by their smallness: "Because of the small student-teacher ratio, students are closer to their professors and receive personal attention" and "I like the fact that it [UNCA] is a small campus, the students and teachers are very nice, and there are classes here that you probably would not find anywhere else," in contradistinction to "UNCW [UNC Wilmington, where] most of the lecture classes were very large. I had several hundred people in my Biology class and also in my Algebra class."

Some commented on the liberal arts curriculum, focusing on the curriculum's breadth as well as disciplinary depth ("Usually liberal arts colleges have more core classes than other colleges. I think this is because the college wants us to have a broader view of horizons rather than focus on the one thing that we like") and on the ways in which the curricular offerings are tied together ("Liberal arts colleges tend to look at education holistically, meaning the curriculum is usually intertwined and well-constructed relating certain areas of study") . They acknowledged that tackling this curriculum is not always easy, but should ultimately be rewarding: "Due to liberal arts requirements one must go outside of their comfort zone and take classes in topics that they are not the best at or comfortable with. Although can be cumbersome as a student at times it helps integrate all areas of study to further the student's knowledge in general."

What are the rewards? Employability, one would hope: "When I graduate I think all those requirements [clusters, writing intensives, etc.] will make me a more rounded person and make me look better to an employer" and "...a degree from a liberal arts school is better to find a job that is not specific, because it shows you have skills in many areas." There's also something to be said for the recognition of the "human" in human scholarly endeavors: "The liberal arts environment at the school has had a major impact on the way I am taught and how I learn here at the University of North Carolina Asheville. [...For example, the Newton v. Leibniz project in Calc I] seemed so absurd but it really helped me realize that the topics we were being taught and the rules that we were learning were not just things that appeared out of thin air. Every subject, every theory, and every idea has a back story, a history, a time, a place, and a brilliant mind who thought it up."

Several students expressed (sometimes extreme) satisfaction at attending a liberals university (a few have spent at least a term in a non-liberal arts environment). Some even expressed concern for their peers elsewhere: "I see a liberal arts education to be very important because I have noticed a terrifying trend that not many of my peers that attend non-liberal arts universities are very secluded and "protected" from the world and the culture in which they live."

Saturday, February 12, 2011

It can be argued that the most important goal of a good university faculty member is to guide successfully her or his students through their development as scholars, specifically as practitioners of one another field and more generally as members of a greater community of rational and reasoning thinkers. (Alternative arguments, many of them cogent, are beside the point of this post.) As such, it's natural that faculty members be critical of their students' work: through careful and constructive criticism we make our and others' expectations clear, and through these means are our students led to understand how to reason and relate with others more clearly.

But as faculty, how critical are we of our own and each others' practices?

I wanted to mention a couple of indirect encounters with Colleagues (a capitalized universal subcommmunity of fellow academics) I had yesterday which really got my goat.

First. Not to harp on this (I know I bitched about it last year, too), but, Colleagues, how in the world can you sleep at night knowing how lousy are the recommendation letters you write on behalf of your students? I'm thinking in particular of a rec letter for an REU applicant I received yesterday which was roughly five lines long, clearly typed directly into the email, and sent all at once (with generic head and foot) to directors of all of the programs to which the student was applying, without even using blind carbon-copy. Never mind the utter disregard for standard email courtesy: five lines is barely long enough, in my experience, to give the merest context, including the student's name, the course(s) in which she enrolled with the writer, and her baseline performance in those courses, to say nothing of individualized evaluation including (a) day-to-day performance in class (was she outgoing, creative, clever, computationally fluent, quick, friendly, supportive, etc.?), (b) her academic work outside of class (did she take on elective projects, undergraduate research, etc.?), (c) her ability to work with others (was she a joiner, a leader, a follower, a groupie, popular with her peers and professors alike?), and (d) her writing ability (which is likely superior to your own)?

Letters like this one embarrass me on behalf of my entire profession. I pity students whose teachers show this little concern for their careers. Sending such a letter says this, and says it loudly: "you are worth roughly ten minutes of my time. I either don't remember you well enough or don't care enough for you to compose a genuine letter of support. I don't really care whether you get into these programs or not. I have better things to do." These letters usually come from faculty at schools like Princeton, Stanford, and Harvard, from faculty who like to think that they're the best in the world at what they do...they may be right, if "what they do" is taken to mean straight-up academic research. They're dead wrong if you interpret their function more broadly and take it to mean enriching the lives of their students, colleagues, and communities (academic and non).

There is a silver lining, of sorts: letters like this one make me realize why it is that my students have a strong track record of getting into graduate programs and REUs. For even the weaker students who ask letters of me I take two or more hours to write a decent letter, almost never under a page in length. Quantity is not quality, but clearly I can and do say much more in fifty lines than I could in five.

Second. On a somewhat related note, I had occasion to read a couple of a colleague's syllabi, and was deeply chagrined by their brevity and wanness. Less than a page long, they did little more than lay our the most basic mechanics of the course: professor's name and contact info, course website, texts, meeting times, grading scheme (bare-bones and anachronistic), and an obligatory statement regarding writing intensiveness and other disclaimers required by the Office of Academic Affairs and the Faculty Senate.

Missing was any kind of statement of purpose, articulation of learning goals, context, description of activities...and in the absence of this matter, the faculty member in question conveyed (as did the letter-writers above) his disdain for his students. Allow me to read (uncharacteristically pessimistically) between the lines: "I don't really care enough about this course to make it clear what we'll do together, or how what we'll do will enrich your experience at this school and provide you help as you grow as a scholar. On most days I'll likely make up class as we go along, because I consider myself too intelligent to have to do much preparation: I can do it on the fly. I'll spend most of my time working on my latest book. If you're outgoing and eager and seek out your own learning opportunities, you'll come by my office and I'll talk to you as I talk to my colleagues (silently applauding myself for empowering you by treating you as a peer and equal), while if you're not so outgoing or eager, however much you may wish to learn in our course, you'll suffer through the semester without any guidance or direction, you'll learn next to nothing, and I'll reward you with a B at the end of the term simply for not bothering me and so that you don't call on me to defend the lousy grade I've given you once the semester's done."

Sound like anyone you know? The kicker: more than once I've heard this colleague help up as a model for excellent teaching.

All right, enough mewling. On to grading. I'll check in later, after I've had another crack at Chapter 1 (a first draft of which I hope...optimistically!...to finish this weekend).

Friday, February 11, 2011

Today was that funnest day of the semester in MATH 280, when everyone gets to take a trip back to the third grade as we play with L-tiles in order to understand the idea of induction. This day is always a fun one, but I don't think any 280 class has ever had as much fun as these folks did. They loved it!

Thursday, February 10, 2011

Despite the late start (again!), yesterday was a full day, pedagogically speaking.

Things got underway in earnest with presentations by my 179 students, on the campus offices and organizations they were to investigate and report upon. Though I've not yet looked at them closely, the brochures they came up with to "advertise" these offices' services look quite good. They're creative and colorful, and some of them almost look professionally done. The presentations showed signs of nervousness and trepidation, which is natural with students at this point in their careers. As I always remind them, getting up and speaking in front of other is hard, no matter how often or how many times you do it. I admired both their willingness to get up and speak and the respect they showed one another as audience members.

I am not, on the other hand, admiring their highly imperfect attendance. Though there's a core of students (15 or so of the 21 now registered for the course) who come unfailingly, the remaining students come only to one out of every three or four class meetings. I've never seen such nonchalance in my courses before.

It's easy to come up with reasons for this. Even I have to admit that the course's subject may not be one which is inherently engaging for students not already interested in math, so many students may find it hard to get excited about the course for its own sake. Since mine is the only 179 course running this semester, several students are in it simply because they have to be, and no one wants to do something on compulsion alone. Finally, 179 courses in general (not just mine) are sometimes seen as "throwaway" courses that students expect to do well in with minimal effort. This is in part because many instructors fail to motivate the class by making its purposes clear.

I hope that in this I'm succeeding. I'm trying to strike a balance between writing instruction (to meet the course's WI goal), overview of the campus and its surrounding community, and actual course "content." It's a precarious balance, and sometimes I feel as though I'm making it up as I go along, and this worries me. However, several activities have seemed to work well ("Everything you always wanted to know about UNCA..." was well-received, and the students seemed to have fun making their meddos). I hope it proves meaningful in the end.

Speaking of public speaking...I ended my workday at Mars Hill College, a small school about 20 minutes north of here. The director of their nascent writing program had invited me to come and lead a workshop for the faculty who are spearheading efforts to get their writing-intensive course program off the ground. There was good disciplinary variety in the participant pool, with social work, journalism, history, literature and composition, religion, and philosophy well-represented. Given that Mars Hill can't possible have more than a hundred or so faculty members, I'm guessing that I met with something like 15% of the school's faculty yesterday. However, I must admit that I was a little chagrined, as were several of the attendants, that the sciences didn't send any of their folks over. (There's talk of me coming back to work with them specifically.)

There was energy in the room: given that Mars Hill's QEP is focused on writing and information literacy, there's strong institutional buy-in for their new WAC program, which features phased-in writing-intensive courses and student writing-fellows. The campus is outwardly stoked about the QEP: banners hang from lampposts exhorting "Write! QEP 2010-2013." It's clearly not just a "top-down" mandate, though; the people I met with yesterday were very much on board with the idea of writing across the curriculum. Their energy was authentic and intrinsic, not simply something generated by an administrative fiat.

Owing to this energy, the workshop was a fruitful one, with active (sometimes even fervent, but always friendly) discussion and idea-sharing. We talked about identifying roles for writing, coming up with learning goals writing could help to meet, structuring writing, using low-stakes writing activities, and responding to writing (while guiding students to do the same, through peer review). We packed a lot into two hours! Judging from the exit slips I received from the participants at the end (thanks for this idea, Libby! I'm going to do this from now on), most people were most interested in learning more about low-stakes writing and its uses in the classroom. The most interesting were the technology-driven (and very game-like) "constrained" forms like tweets and texts, as well as my newly-minted "Intrigue, Confusion, and Confidence."

I picked up some ideas, too. One of the conference participants offered a low-stakes writing activity he's used to help his shyer students get their say in in-class discussions. Rather than requiring them to speak up in front of a couple dozen of their peers, he asks students who are hesitant about contributing talking points to email their ideas ahead of time so that he can collect these ideas and share them with the class on the overhead. Those students' discussion points are on display, anonymously, when the class comes in, and in this way even the less outgoing students are able to make their voices heard. I asked if this method of starting the class conversation has helped the shy students come out of their shells (as, for instance, they might be called on to defend the points they've made via email), and he indicated that this is the case.

At the end of the workshop I had a chance to dish about the reception on the part of faculty of UNCA's own WI program. I said that I think it's generally good, and that though there was resistance when ILS was first phased in several years ago, most of the younger faculty have bought into it authentically and this has led to a campus culture in which writing-intensive courses are broadly accepted. I honestly think that this is the case: there are grumblers and groaners, but for the most part the mission of the intensives is unquestioned, and teaching WI courses is a valued activity.

Late last night, after hammering away at Chapter 1 of my book for several hours straight, I mused that writing may be "frozen thought," in the same way Goethe referred to architecture as "frozen music." Late it was, indeed: I notice things now, in matinal lucidity, that I may not have last night.

For instance, I ought not simply to say "writing," but rather, less aphoristically but more appropriately, "the product of our writing." I of all people should know not to equate that which is written with the process by means of which that "that" is produced. And that process, "writing" in the true sense, is hardly frozen; instead, it is dynamic and fluid. It is a frozen river thawed.

Rather than criticize my own pith, however, I suppose I could level a charge against Goethe's parallel antecedent. Surely any architect worth her salt, even in Goethe's time, would grimace to equate the building with the built?

Wednesday, February 09, 2011

I've gotten to the point where I don't think I can think unless I'm writing my thoughts out.

"Write, write, write," I exhort my students. "Write out your thoughts, in whatever form they take." "Play!" is a variant of this exhortation.

It's taken me over a decade and a half to realize that what I call "play," and what my wonderful mentor Jim Hagler no doubt called by this same word when he encouraged me with it during my own years as an undergraduate, is simply a mathematical version of low-stakes writing. "Play" can comprise words and numbers and notation, but it can also comprise visual elements such as pictures, charts, and graphs.

What purpose does "play" serve? The same purpose served by any writing-to-learn activity. I reminded my 280 students this morning that the very act of rendering concrete the abstract contents of our convoluted brains can help us make sense of those contents. The recursive processes involved in "play" (drawing, drafting, doodling, then reflecting, recollecting, and redrafting) help us organize our thoughts in ways we never could if we kept those thoughts confined.

Case in point? This afternoon I was working with one of the three math students whom I'm guiding in undergraduate research this term. Lulu's a terrifically intelligent student with whom I had the pleasure of working two years ago this term when she enrolled in my 280 course. She was easily one of the top two or three students in that large class, and since then she's distinguished herself by continued success in several upper-level math courses. Right now she's doing original work with me on the Möbius function of various subgroup and subgraph lattices, and she's making great progress so far. (Damn it...I've written so damned many rec letters this season that my blog posts are starting to sound like one...)

Today I was introducing her to the Deletion-Contraction algorithm for the recursive computation of chromatic polynomials. To illustrate the algorithm, I thought to use it to compute the chromatic polynomial of the cycle on 6 vertices, C6. We began our work playfully, tinkering with the graphs which resulted from applying the algorithm to the initial graph. It soon became evident (through the simple stick charts which we drew) that we'd only find our answer if we knew something about the chromatic polynomial of P6, the path on 6 vertices.

We let play push us forward. P6, itself not the initial object of our investigation, soon led us further astray, as P6 gave way to P5, and this to P4, and so on. A few computations (and a hand-wavy inductive argument) later, and we had a formula for the chromatic polynomial of Pn.

Play pushed us back, back to C6. P6 subdued, we turned to C5, also intimately involved. But analogy (again recognized by the stick charts we'd drawn) allowed us to develop a pattern for this graph, too, and we soon realized a sort of inclusion/exclusion formula for the chromatic polynomial we sought.

Whether or not you follow the technical details above, I hope you can recognize this: the tinkering and tweaking we did on the blackboard in my office, including both playful formulaic experimentation and purposeful doodling, is precisely what helped us to develop the formula we sought. There is simply no way we could have arrived at that formula had we simply stared into space and "thought about" the graph C6. Though the writing I was doing as Lulu and I worked this problem together helped me to communicate some new mathematical ideas to my young mentee, the writing also served the more critical role of making real my once-cluttered/now-clear thoughts.

Lately in all of my classes I've been using a variation on the classic "three-minute theme" low-stakes writing activity in which I ask students to write three things about a reading or a class discussion or a course in its entirety:

1. What one topic are you most intrigued about, or want to know more about, right now?

2. What one topic are you most confused about right now?

3. What one topic are you most confident about right now?

Each of these prompts has a particular purpose. The first helps students identify their interests and better understand why it is they might find our coursework appealing. This interest and appeal translates into motivation to keep working at the course. The third prompt has much the same effect: by reflecting on a topic she feels confident about, a student is likely to say to herself "yeah, I can get this! I'll keep at it, and sometime soon the rest will come just as easily." Meanwhile in the reflection needed to respond to the second prompt the student will identify areas on which she may need to focus extra effort.

I'm calling this activity "Intrigue, Confusion, and Confidence," for obvious reasons. I've already used it several times this term to help my MATH 179 class focus their discussions, and I just used it a half-hour ago to help me figure out where it is my Calc II students are feeling good about themselves...and where it is they might need extra work.

It's worth noting that we've just today wrapped up our discussion of numerical integration, so those topics may appear more frequently because they're fresh in mind. They're also not quite as complicated conceptually as some of the other topics we've covered. I find it interesting that as many people are fine with volumes as are iffy about them; this complication makes it difficult to give any specific prescription for study or review. I'll put together some additional practice problems for people to tackle tomorrow after the quiz.

If you're one of the students in my Calc II class and you're reading this, do me a favor: take five minutes to go to the comments section and let me know if this brief exercise at the end of class today was helpful to you...and if so, why. I appreciate it!

By the way, the two most entertaining comments, both about areas of confidence: "Rotating functions around axes is SUPER FUN! [accompanied by a Bundt-pan shaped volume of revolution]" and "That thingy. Can't think of process name. [accompanied by a sketch of a vase-like object suggesting a disk-method volume-finding question]"

Tuesday, February 08, 2011

We had some great conversations in my Ethnomathematics course today, a day we devoted to trying to obtain a better and broader view of the university we all work at. After doing a peer review of each others' brochures (advertising campus offices and organizations of important to new arrivals to campus), I asked the students what it was that drew them to UNC Asheville in the first place.

The list they came up with was a comprehensive one. According to them, the draw of the school has to do with the fact that UNCA

is cheap,

is in the mountains,

is in North Carolina,

is in Asheville,

is close to family,

offers a "real" [I think he meant "more comprehensive"] education, as opposed to a conservatory,

has a small campus,

has a green campus,

has promising programs of interest [teaching licensure and Chemistry were mentioned specifically],

isn't pricey,

has nice faculty,

has an accepting community,

is part of an active larger community,

is situated in a "hippie town,"

has no football team,

offers diversity,

offers a large number of extra-curricular activities,

offers small classes,

allows students flexibility in their studies, and

offers a broad variety of courses one wouldn't find elsewhere.

I pressed the students a bit on this last point: what courses do you think distinguish UNCA from other institutions?

"This one," one student offered. "This doesn't seem like the kind of class you'd be able to take just anywhere." Students also cited a number of other courses, many of which have an interdisciplinary flavor to them: a jazz course on the popularity of The Beatles, physics courses on music and sound, Asian studies courses, and teaching licensure courses relating to every discipline across campus.

Toward the end of our conversation, we began to drift toward the topic on which I asked the students to write a brief reflection for next week: what is it about UNCA that makes it a liberal arts institution? Judging from their responses to this question, I think the students have a good idea already, and I'm looking forward to further discussion:

"I went to UT [University of Tennessee] before this, and there all of the teaching was straight out of the textbook. You had to read the text and then you'd get lectured to about it. Here there's so much more variety in the way things are taught. And the classes all fit together and build on one another. They affect one another."

"There are connections between all of the different components of our education."

"We have a chance to learn in small classes like this one, instead of large lectures with several hundred people."

"The range of courses we can take is huge. And they sometimes consider the same ideas from different points of view."

Unwittingly the students are hitting some of the reasons I came to UNCA and enjoy my career here so much: I have the chance to actually get to know my students and their hopes and dreams. I have the chance to teach rich courses with a similarly rich assortment of pedagogical techniques. Moreover, I do this all with strong students who would never be able to afford the cognate experience at a private school like Bucknell, Bard, or Davidson. It's a good life.

I'm looking forward to reading the students' reflections...and to their presentations on the campus offices to which they've been assigned. More to come!

Monday, February 07, 2011

At the end of Calc II today I asked the students to write two things on a scrap of paper: (1) whether they'd like to get the first take-home exam this Friday or next Friday and (2) on a scale of 1 to 10, how they feel about our class so far (with additional commentary as needed).

Most of the students are really enjoying the class and getting a lot out of it, giving a rating of either 9 or 10, and in one case 11, for the second question. (There were, of course, some oddballs who insisted on giving a rating of 3π or √97 or some similar silliness.) Most people loved the way the class is set up, with a high level of interaction, applicability, challenging but authentic problems, and fun. "I'm enjoying the togetherness," said one student, and another said "I really enjoy your approach. It can be overwhelming, and a lot of work. But you make it fun."

The most common concern was the speedy pace of the class: 4 out of 30 students mentioned this concern. (3 out of 30 mentioned the amount of work, the next-most-common concern.) The pace is, sadly, hard to make much slower, given how much mileage we're expected to make before Calc III. (And, as I point out almost every term, I make my way through these topics more slowly than any of my colleagues do. I know my department chair's section of Calc II is about three sections ahead of us right now.)

There are ways to make the pace and workload feel less stressful:

Do you feel like you're having a hard time keeping up with notes in class? Come to class with my "skeletal" notes already printed out: you'll have less to write and can focus more on the ideas. I strongly recommend this route. I know it's helped a lot of people in previous semesters.

Do you feel overwhelmed by the homework load? Get started on the homework as soon as it's assigned, and identify immediately any potential problems you think you might encounter. Ask me about these problems right away, so that you're not stuck asking them on Friday afternoon, an hour before the homework is due. (Things always feel more stressful right before the deadline.)

Do you feel like you're getting the ideas as we talk about them, but that they're just not sinking in as well as you'd like them to? Talk them out with your friends in the class. Draw lots of pictures as you explain the ideas to your friends. Puzzle through the pictures together and make sure you can tie the conceptual ideas to the computations that go hand-in-hand with them. Not only do your friends sometimes do a better job than I do in explaining these concepts; moreover often the best way to learn something is to teach it to someone else. (I honestly didn't truly understand Calc II until I started teaching it as a graduate student.)

Above all else, remember that you're not alone. We have an overwhelmingly friendly class, and I've seen how wonderfully you're all working together. Take advantage of that sense of togetherness and help each other out. You're making a great team.

Oh, and this Friday beat out next Friday, 17 to 12 (with 1 abstention). I believe I will pass out the exam this week, but please recall that you'll have a week in which to do it.

Friday, February 04, 2011

My first visit to East Carolina University (ECU) has been a successful one so far, at least from my point of view. (You can ask my interlocutors in the Math Department and the university’s current WAC [Writing Across the Curriculum] Academy if they feel otherwise.) I’ve been here for a day and a half now, and it’s been pretty much nonstop action.

I got into town on Wednesday night around 7:30, having left Asheville just minutes after wrapping up my early-afternoon Calc II class. There was still enough time left on the evening to catch a late dinner at Copper & Vine with my wonderful colleagues and hosts Libby and Xavier (holla!), faculty in the ECU Writing Center.

Yesterday morning gave me some free time to finish off a couple more rec letters before the real work began. Over coffee and tea at Starbucks Libby and I went over the feedback she’d written on Chapters 3 and 6 of my book. She seemed to be apologetic about offering her comments with an eye toward reader response, but I assured her that this is exactly what I need. Besides pointing out potential points of confusion, her comments offered up several sources with which I wasn’t familiar before, and easily a dozen new ideas for low-stakes writing activities, methods of providing feedback on writing, and means of making the writing process smoother and more meaningful for student writers.

We bustled off to lunch at the Starlight Café (if nothing else when this trip is done I’ll able to tell you where to eat in Greenville) with a few more folks from the Writing Center, and shortly thereafter it was time for my first performance, a research talk in the Mathematics Department.

I was well-received and my talk was somewhat well-attended…but somehow the energy wasn’t there for me. It’s no secret to people close to me that I’ve recently found myself a little disillusioned by mathematics: there’s little sense of urgency to it, there’s little authenticity to what I do. This isn’t to say that math is pointless and purposeless; it’s simply that lately I fear I find little purpose in it for myself beyond play-like manipulation or action-under-constraint, a sort of glass-bead game or extended Oulipian exercise. I fully recognize that the mathematics I do has little, if any, immediate impact on the world and I feel that on the other hand most other aspects of my career and personal pursuits and passions (teaching, writing, and poetry) have relatively profound impact.

Couple this with the fact that mathematicians are, let’s face it, not the most outgoing or socially “ept” of individuals by and large, and I feel as though I’m alienating myself from my own discipline.

Meanwhile I’m moving closer and closer to another: right now teaching writing to me seems a much more personal, passionate, fulfilling, rich, and worthwhile activity. While one in ten thousand people might need to know what the independence polynomial of a path-like graph tells us about that graph’s connectivity, it’s impossible to know too well how best to express oneself. It doesn’t hurt that nearly every one of my colleagues in composition and rhetoric is warm and welcoming. I’m sure the discipline’s got trolls of its own; it may just be that they do a better job of hiding them.

Straight from my math talk I walked across the quad with my Quimby (another of the ECU Writing Center folks…both he and Libby attended my talk in the Math Department, brave souls!) to yesterday evening’s meeting of the WAC Academy. This six-week faculty development series challenges its participants to find more meaningful ways to involve writing in their disciplinary courses. Perhaps unsurprisingly (this is always the case in such workshops) there is a preponderance of younger faculty involved in this program, but the various disciplines are broadly represented in this semester’s group, spanning from psychology and biology at one end to drama and dance at the other.

The focus of the evening was The Writing Process, reverently rendered in capitals. Libby (who just now stopped by where I’m sitting as I write this sentence), the WAC Academy’s fearless leader, started things off by leading a “writing into the day” activity. She asked participants (and me) to illustrate the process they performed when writing their most recent substantial writing project. I portrayed my book-writing project as an Iron Chef-like jumble of too many skillets on too many stoves which gradually became more like an ever-revolving Ferris wheel, one chapter on each of several gondolas that got written on each time it spun past the spot where I stood on the ground. It was an intentionally playful exercise that definitely helped me to pinpoint where and when it is I do my best writing, and what I need to perform that writing.

Unsurprisingly, several others’ writing portraits belied decidedly different processes. One writer combined passion and intellectualization in a picture of a brain with giant red lips: a “brainkiss.” (This is the same man whose list of New Year’s resolutions included the perfection of the “cloud hug.”) A scholar in child development portrayed her writing of a grant proposal as a regular succession of polygons, signifying a step-by-step process which began with an idea and ultimately ends by branching into new ideas (represented by the same circle which symbolize the initial idea). Cyclic and circular, linear and direct, all manner of processes were represented. Surely this variety in the final product is expected, and the “discovery” of such manifold means of writing is one of the take-home points of the exercise.

My portion of the evening’s activities gave the participants a taste of the textbook writing exercise I’ve used in MATH 280 and, with less success, in Topology. (I find it interesting that as I type this sentence my 280 students are hard at work, I hope, in outlining the first chapter! I just texted Iris to see if all is well there.) I asked the participants to brainstorm ideas for the first “chapter” of a text on the ideas they’ve encountered in the first week and a half of the Academy, select five topics on which to base “sections” for the text, and then in pairs write two or three sentences on each of those topics. The emphasis through and through was on the underlying process. Though we didn’t have time to review and revise the writing they’d done, I believe the exercise was well-received and helpful.

As always, the informal and after-hours activities provided as meaningful a learning experience as the formal ones. Dinner at Wasabi 88 gave me a chance to swap traveler’s tales of disciplinary dragons, budgetary woes, and ideas for potential future collaboration. Several of my writing colleagues came, as did three of the brand new ECU faculty. Boudica, Irving, and Monica talked about the frustrations and felicitations they’ve felt during their first year at a new school. They dream big, talking about cross-listed courses they’d like to teach and plans they have for using collaborative learning and boundary-blurring pedagogies in their classes. (Think “theater of the oppressed” and “literature of the wilderness.”) The future is in good hands.

After dinner I spent a couple more hours putting together materials for my final performance at ECU, a presentation I’ll be giving in about two hours to the ECU Writing Intensive faculty. I’ll be hitting the highlights of writing-to-learn, indicating several useful types of WTL activities and letting participants play with a few of them. (I’m toying with the idea of assigning a lipogram…)

It’s been a busy, but bountiful, trip. I’ll be glad to be home, and I’ll have returned a richer man.

Tuesday, February 01, 2011

Today's MATH 179 class was the most fun yet...at least for me. I felt both major activities went pretty well, and I finally get the sense that the students in that class are opening up to and relaxing around one another, and that they're acknowledging and understanding the sort of work that I'm expecting of them.

Our reading today treated three examples of calendars that reconcile lunar cycles with solar cycles (including the very hard-to-understand Jewish calendar). As an entry into discussing the calendars, I used a low-stakes writing activity I've been developing for a bit now: "Intrigue, Confusion, and Confidence." I ask each student to write three sentences: in the first the student indicates something in the reading which intrigued or excited her; in the second she indicates something she was confused about; in the third she indicates something she feels confident she could explain well to someone else.

After writing for five minutes, I have the students pair off and compare their lists. I ask them to fill one another in if one is confused about something the other knows confidently. I ask them to come up with at least one point they'd like to bring up in discussion.

Five more minutes later, we come together as a class and compare the things we've all come up with which are intriguing, confusing, or inspiring of confidence. Students are heartened to see how similar their lists often are: they all have trouble on the same things. (Like the Jewish calendar!)

The second half of class was devoted to an activity I called "Everything you always wanted to know about UNCA (but were afraid to ask)." I passed around identical index cards and asked each student to write down any questions they had about our school to which they wanted to find answers. "I know how it can be embarrassing asking your friends or professors. Some questions seem like silly ones, and you don't want people to think 'he doesn't know that!?' " After I collected the cards, I read them aloud, one by one, pausing so that anyone in the room who had a response could pose an answer.

The ensuing discussion was fantastic. Two or three of the students proved to be particularly knowledgeable about our campus, and I feel they'll make excellent mentors as the semester goes by. It was an empowering conversation, I felt:

1. the students realized lightning bolts would not smite them for asking questions;

2. the students realized that their peers had many of the same questions they did (thereby making those questions by definition "not dumb"); and

3. the students realized that they personally often had the answers to questions their peers might pose, making them the authorities.

I'll definitely repeat this exercise several times this semester, and I'll probably use it in other courses, about course content, too.